| L(s) = 1 | − 18.3·3-s − 72.1·5-s + 96.4·7-s + 93.5·9-s + 684.·11-s + 344.·13-s + 1.32e3·15-s + 1.31e3·17-s − 739.·19-s − 1.76e3·21-s − 3.16e3·23-s + 2.08e3·25-s + 2.74e3·27-s + 7.07e3·29-s + 3.79e3·31-s − 1.25e4·33-s − 6.96e3·35-s − 1.26e4·37-s − 6.32e3·39-s − 1.08e4·41-s − 1.84e3·43-s − 6.74e3·45-s − 3.87e3·47-s − 7.49e3·49-s − 2.42e4·51-s + 6.47e3·53-s − 4.93e4·55-s + ⋯ |
| L(s) = 1 | − 1.17·3-s − 1.29·5-s + 0.744·7-s + 0.384·9-s + 1.70·11-s + 0.565·13-s + 1.51·15-s + 1.10·17-s − 0.469·19-s − 0.875·21-s − 1.24·23-s + 0.667·25-s + 0.723·27-s + 1.56·29-s + 0.708·31-s − 2.00·33-s − 0.961·35-s − 1.52·37-s − 0.665·39-s − 1.01·41-s − 0.152·43-s − 0.496·45-s − 0.255·47-s − 0.446·49-s − 1.30·51-s + 0.316·53-s − 2.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.176318410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.176318410\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 3 | \( 1 + 18.3T + 243T^{2} \) |
| 5 | \( 1 + 72.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 96.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 684.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 344.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.31e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 739.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.16e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.08e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 3.87e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.47e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.47e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.25e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.931238686519774360434597460002, −8.526399417813880939989135879919, −8.122037326884990971094931619014, −6.85841183050208225973004207145, −6.25332133180993091503410394407, −5.12582737572426745820098666525, −4.23934016267003454534576670225, −3.47074249934052417185515878539, −1.49708120032659719824150272545, −0.59356792320131411206399599912,
0.59356792320131411206399599912, 1.49708120032659719824150272545, 3.47074249934052417185515878539, 4.23934016267003454534576670225, 5.12582737572426745820098666525, 6.25332133180993091503410394407, 6.85841183050208225973004207145, 8.122037326884990971094931619014, 8.526399417813880939989135879919, 9.931238686519774360434597460002
